Is it possible to record the bevel factor as a keyframe in the action editor?
it's because bevel depth is not an object property - it's curve data property. Action editor shows only one animation block (bordered with red line):



Outer animation blocks doesn't display in action editor

So what you can do - create a property in object data and make driver to manipulate bevel depth value in curve data 
through object data. 

Howto:


In object data find tab custom properties  and add one:





Next, open curve data side by side and press RMB on bevel depth:




Chose add driver -> single from target, and (when your cursor will be as dropper) click on value created before.


So now you have value in action editor, that manipulates bevel depth: